Electron occupations that arise from pure quantum states are restricted by a stringent set of conditions that are said to generalize the Pauli exclusion principle. These generalized Pauli constraints (GPCs) define the boundary of the set of one-electron reduced density matrices (1-RDMs) that are derivable from at least one N-electron wavefunction. In this paper, we investigate the sparsity of the Slater-determinant representation of the wavefunction that is a necessary, albeit not sufficient, condition for its 1-RDM to lie on the boundary of the set of pure N-representable 1-RDMs or in other words saturate one of the GPCs. The sparse wavefunction, we show, is exact not only for 3 electrons in 6 orbitals but also for 3 electrons in 8 orbitals. For larger numbers of electrons and/or orbitals in the lowest spin state, the exact wavefunction does not generally saturate one of the GPCs, and hence, the sparse representation is typically an approximation. Because the sparsity of the wavefunction is a necessary but not sufficient condition for saturation of one of the GPCs, optimization of the sparse wavefunction Ansatz to minimize the ground-state energy does not necessarily produce a wavefunction whose 1-RDM exactly saturates one of the GPCs. While the sparse Ansatz can be employed with arbitrary orbitals or optimized orbitals, in this paper, we explore the Ansatz with the natural orbitals from full configuration interaction, which yields an upper bound to the ground-state energy that equals the exact energy for a given basis set if the full-configuration-interaction wavefunction saturates the Ansatz's GPC. With calculations on the boron isoelectronic sequence, the dinitrogen cation N2+, hydrogen chains, and cyclic conjugated π systems, we examine the quality of the sparse wavefunction Ansatz from the amount of correlation energy recovered.